Bunuel wrote:
If F is the prime factorization of N!, how many factors in F have an exponent of 1?
(1) 30 ≤ N ≤ 40
(2) 25 ≤ N ≤ 35
Kudos for a correct solution.
VERITAS PREP OFFICIAL SOLUTION:This is a tricky one.
Let’s think about, say, 40! This number, 40!, is the product of all the integers from one to 40. Let’s think about its prime factorization. It would have at least one factor of 2 for every even number from 2 to 40, and a second factor for every multiple of 4, and a third factor for every multiple of 8, etc.; a lot of factors of two. Think about the factors, say, of 7: there are five multiples of 7 from 7 to 35, so in the prime factorization of 40!, the factor 7 would have an exponent of 5. Which factors would have exponents of 1? Well, the prime numbers that are less than N, but have no other multiples less than N. For example, in 40!, the factor 37 would have an exponent of 1 since it appears once and no other multiple of it is less than 40.
Statement #1: 32 ≤ N ≤ 40
As we move through different N’s in this region, we cross the prime number 37, which will have an exponent of 1 if it appears. Some N’s include this prime number and some don’t, so the number of factors with an exponent of 1 is different for different values of N. This statement, alone and by itself, is not sufficient.
Statement #2: 27 ≤ N ≤ 35
As we move through different N’s in this region, we cross two prime numbers, 29 and 31, each of which will have an exponent of 1 if it appears. Some N’s include neither, some include 29 and not 31, and some include both, so the number of factors with an exponent of 1 is different for different values of N. This statement, alone and by itself, is not sufficient.
Combined: 32 ≤ N ≤ 35
Now, there are no prime values in the range specified. But, here’s a tricky thing. If N = 32 or 33, then either 32! or 33! contains exactly one factor of the prime numbers {17, 23, 29, 31}: four prime factors with an exponent of one. BUT, if N = 34 or 35, there are now two factors of 17 (one from 17 and one from 34), either 34! or 35! contains exactly one factor of the prime numbers {23, 29, 31}: three prime factors with an exponent of one. Even in this narrow range, different choices lead to different answers for the prompt question. Even together, the statements are not sufficient.
Answer = (E)
Why 19 is missing from the set of exactly one factor of the prime numbers i.e. {17, 23, 29, 31} ?